یک کاربرد مؤثر از کنترل بهینه کسری در درمان بیماری‌های عفونی
محورهای موضوعی : مهندسی برق و کامپیوترامین جاجرمی 1 * , منیژه حسن آبادی 2
1 - گروه مهندسی برق، دانشکده فنی و مهندسی، دانشگاه بجنورد، ایران
2 - گروه ریاضیات کاربردی، دانشکده علوم ریاضی و آمار، دانشگاه بیرجند، ایران
کلید واژه: دینامیک انتقال عفونت HIV, کنترل بهینه, مشتقات کسری, نتایج عددی,
چکیده مقاله :
در این مقاله، یک مدل کنترل بهینه جدید به کمک نسخه تعمیم‌یافته مشتق کسری با تابع کرنل دلخواه برای بررسی و کنترل بیماری عفونی HIV پیشنهاد می‌شود. بدین طریق یک استراتژی درمانی کارآمد بر اساس نظریه کنترل بهینه کسری برای بیماری ذکرشده تدوین و بررسی می‌شود. شرایط لازم بهینگی بر اساس اصل ماکسیمم‌یابی پونتریاگین به فرم یک مسئله مقدار مرزی کسری فرموله می‌شود. علاوه بر این، یک روش عددی کارا و مؤثر برای یافتن پاسخ سیستم دینامیکی کسری تعمیم‌یافته و همچنین حل مسئله کنترل بهینه متناظر با آن ارائه می‌شود. همگرایی و تحلیل خطای تکنیک پیشنهادی نیز مورد مطالعه و بررسی قرار می‌گیرد. شبیه‌سازی‌های عددی نشان می‌دهند که مدل کسری تعمیم‌یافته برای مرتبه کسری 99/0 دارای خطای کمتر و در نتیجه دقت بهتر در مقایسه با مدل مرتبه صحیح و نیز سایر مراتب کسری آزمایش‌شده (بر اساس داده‌های واقعی تأییدشده توسط سازمان بهداشت جهانی) می‌باشد. به‌علاوه اعمال کنترل بهینه پیشنهادی منجر به کاهش قابل توجهی در گسترش بیماری می‌شود؛ در نتیجه مدل کسری معرفی‌شده ابزاری کارآمد برای نشان‌دادن ویژگی‌های اساسی انتقال بیماری مفروض بوده و می‌تواند باعث افزایش کارایی استراتژی‌های درمانی شود.
In this paper, a novel optimal control model is proposed using the generalized fractional derivative with an arbitrary kernel function to investigate and control HIV infection. Accordingly, an effective therapeutic strategy based on the fractional optimal control theory is formulated and examined for the mentioned disease. The optimality conditions, based on the principle of Pontryagin’s maximization, are formulated as a fractional boundary value problem. Furthermore, an efficient numerical method is presented for finding the solution to the generalized fractional dynamic system and solving the corresponding optimal control problem. Convergence and analysis of the proposed technique’s error are also studied and evaluated. Numerical simulations demonstrate that the fractional derivative model for a fractional order of 0.99 exhibits lower error and, consequently, better accuracy compared to the integer order model and other fractional orders tested (based on real data validated by the World Health Organization). Moreover, the implementation of the proposed optimal control leads to a significant reduction in the spread of the disease. Consequently, the introduced fractional model serves as an efficient tool for highlighting the fundamental characteristics of the transmission of the aforementioned disease and can enhance the effectiveness of therapeutic strategies.
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